parrevgum {lmomco} | R Documentation

## Estimate the Parameters of the Reverse Gumbel Distribution

### Description

This function estimates the parameters of the Reverse Gumbel distribution given the type-B L-moments of the data in an L-moment object such as that returned by `pwmRC` using `pwm2lmom`. This distribution is important in the analysis of censored data. It is the distribution of a logarithmically transformed two-parameter Weibull distribution. The relation between distribution parameters and L-moments is

α = λ^B_2/lbracelog(2) + mathrm{Ei}(-2log(1-zeta)) - mathrm{Ei}(-log(1-zeta))rbracembox{ and}

xi = λ^B_1 + αlbracemathrm{Ei}(-log(1-zeta))rbracembox{,}

noindent where zeta is the right-tail censoring fraction of the sample o the nonexceedance probability of the right-tail censoring threshold, and mathrm{Ei}(x) is the exponential integral defined as

mathrm{Ei}(X) = int_X^{infty} x^{-1}e^{-x}mathrm{d}x mbox{,}

noindent where mathrm{Ei}(-log(1-zeta)) rightarrow 0 as zeta rightarrow 1 and mathrm{Ei}(-log(1-zeta)) can not be evaluated as zeta rightarrow 0.

### Usage

    parrevgum(lmom,zeta=1,checklmom=TRUE)

### Arguments

`lmom` | A L-moment object created by `pwm2lmom` through `pwmRC` or other L-moment type object. The user intervention of the `zeta` differentiates this distribution and hence this function from similar parameter estimation functions in the lmomco package.
`zeta` | The right censoring fraction. Number of samples observed (noncensored) divided by the total number of samples.
`checklmom` | Should the `lmom` be checked for validity using the `are.lmom.valid` function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the tau_4 and tau_3 inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.

### Value

An R `list` is returned.

`type` | The type of distribution: `revgum`.
`para` | The parameters of the distribution.
`zeta` | The right censoring fraction. Number of samples observed (noncensored) divided by the total number of samples.
`source` | The source of the parameters: “parrevgum”.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

### See Also

`pwm2lmom`, `pwmRC`, `cdfrevgum`, `quarevgum`

### Examples

    # See p. 553 of Hosking (1995)
    # Data listed in Hosking (1995, table 29.3, p. 553)
    D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443,
           -1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620,
           -0.576, -0.548, -0.247, -0.195, -0.056, -0.013,  0.006,
            0.033,  0.037,  0.046,  0.084,  0.221,  0.245,  0.296)
    D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions 40 values in total
    z <-  pwmRC(D,threshold=.2960001)
    str(z)
    # Hosking reports B-type L-moments for this sample are
    # lamB1 = -.516 and lamB2 = 0.523
    btypelmoms <- pwm2lmom(z$Bbetas)
    # My version of R reports lamB1 = -0.5162 and lamB2 = 0.5218
    str(btypelmoms)
    rg.pars <- parrevgum(btypelmoms,z$zeta)
    str(rg.pars)
    # Hosking reports xi = 0.1636 and alpha = 0.9252 for the sample
    # My version of R reports xi = 0.1635 and alpha = 0.9254

[Package lmomco version 0.92 Index]